Theory of transport through quantum-dot spin valves in the weak-coupling regime

We develop a theory of electron transport through quantum dots that are weakly coupled to ferromagnetic leads. The theory covers both the linear and nonlinear transport regimes, takes noncollinear magnetization of the leads into account, and allows for an externally applied magnetic field. We derive generalized rate equations for the dot’s occupation and accumulated spin and discuss the influence of the dot’s spin on the transmission. A negative differential conductance and a nontrivial dependence of the conductance on the angle between the lead magnetizations are predicted.

Figure 1

Quantum-dot spin valve: a quantum dot is weakly coupled to ferromagnetic leads. The magnetization directions of the leads enclose an angle $\varphi$.

Figure 2

(a) Scheme of the quantum-dot energy scales. The difference of the electrochemical potentials of the left and right leads describes a symmetrically applied bias voltage. (b) The coordinate system we choose. The magnetization directions ${\widehat{\mathbf{n}}}_{\mathrm{L}}$ and ${\widehat{\mathbf{n}}}_{\mathrm{R}}$ enclose an angle $\varphi$.

Figure 3

Graphical representation of the phase factors in the tunnel Hamiltonian, Eq. (2.2). In this representation, the quantum-dot spin valve bears similarities to an Aharonov-Bohm setup, with the different arms modeling the two different spin channels ↑ and ↓, and the Aharonov-Bohm flux corresponding to the relative angle of the leads’ magnetizations.

Figure 4

Spin dynamics in the linear-response regime. Spin accumulates along the $y$ direction. The spin precesses due to the exchange field that is along the $x$ direction. Therefore, the stationary solution of the average spin on the dot is tilted away from the $y$ axis by an angle $\alpha$, plotted in Fig. 5b.

Figure 5

(a) Linear conductance normalized by $\Gamma ∕k_{\mathrm{B}}T$ as a function of the level position $\epsilon$ for different angles $\varphi$. (b) Angle $\alpha$ enclosed by the accumulated spin and the $y$ axis as defined in Fig. 4. (c) Derivative of the magnitude of the accumulated spin on the dot with respect to the source-drain voltage $V$. Further parameters are $p=0.9$ and $U=10k_{\mathrm{B}}T$.

Figure 6

Normalized conductance as a function of the angle $\varphi$ enclosed by the lead magnetization for different level positions and the parameters $U=10k_{\mathrm{B}}T$ and $p=0.9$.

Figure 7

Current-voltage characteristics for antiparallel (a) and perpendicular aligned (b) lead magnetizations. Further parameters are ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}=\Gamma ∕2$, $p_{\mathrm{L}}=p_{\mathrm{R}}=p$, $\epsilon =10k_{\mathrm{B}}T$, and $U=30k_{\mathrm{B}}T$.

Figure 8

For electrons polarized antiparallel to the drain lead, the influence of the effective field generated by the source lead is dominating. By rotating the spins, the spin blockade is lifted and therefore the conductance recovers.

Figure 9

Panel (a): The absolute value of the effective exchange field contributions from the left and right leads. Panel (b): the current voltage dependence, with and without the influence of the exchange field. For both plots the parameters $\varphi =\pi ∕2$, ${\Gamma }_L={\Gamma }_R=\Gamma ∕2$, $\epsilon =10k_{\mathrm{B}}T$, $U=30k_{\mathrm{B}}T$, and $p=0.95$ were chosen.

Figure 10

Angular dependence of the conductance with an applied voltage of $V=\epsilon +U∕2$, i.e., the voltage generating the smallest influence of the exchange field. Further plot parameters are ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}=\Gamma ∕2$, $p_{\mathrm{L}}=p_{\mathrm{R}}=p$, $\epsilon =10k_{\mathrm{B}}T$, and $U=30k_{\mathrm{B}}T$.

Figure 11

Linear conductance as a function of an applied external magnetic field in $x$ direction. The parameters are ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}=\Gamma ∕2$, $p_{\mathrm{L}}=p_{\mathrm{R}}=p$, $\epsilon =0$, $U=10k_{\mathrm{B}}T$, and $\varphi =\pi ∕2$.

Figure 12

Examples for the diagrams representing the generalized transition rates $\sum$.

Figure 13

Examples for the diagrams representing Green’s functions.